Shrink-Fit Reinforced Coil Design and Analysis

Hello! Welcome to the next lesson in our course on high-field magnet design.

In our last session, we explored how distributed reinforcement, particularly with Zylon fibers, allows a coil to withstand immense hoop stresses by having the stronger, stiffer reinforcement material carry a larger portion of the load. We established that for a 60 T field, the composite nature of the winding is essential for survival.

Today, we'll delve into a complementary and equally crucial technique for managing mechanical stress: pre-compression through shrink-fitting. While distributed reinforcement adds strength within the winding, shrink-fitting gives the entire coil a mechanical "head start" against the explosive forces of the pulse. Our learning outcome is to calculate the required interference fit and assembly stresses for a shrink-fit reinforced coil. This involves applying the principles of solid mechanics to a very practical engineering problem central to the construction of robust pulsed magnets.

1. The Principle of Shrink-Fit Reinforcement

A shrink-fit is an elegant solution to a brute-force problem. The idea is to encase the magnet coil in a very strong, thick-walled cylinder (the reinforcement). Crucially, the inner diameter of this reinforcing cylinder is manufactured to be slightly smaller than the outer diameter of the coil.

To assemble this, one of two things happens:

The outer reinforcement is heated, causing it to expand just enough for the coil to be inserted.
The inner coil is cooled (often with liquid nitrogen), causing it to contract so it can fit inside the reinforcement.

Once the components are assembled and return to thermal equilibrium, the reinforcement tries to contract back to its original size, and the coil tries to expand. Since they are interlocked, they can't. This creates a large, permanent contact pressure at the interface.

The result is that the inner coil is placed under a significant compressive stress before the pulse even begins. When the current pulse generates the outward Lorentz force, this force must first overcome the initial compressive stress before it can begin to induce any dangerous tensile stress in the conductor. This dramatically increases the field the coil can withstand before reaching its failure point.

The image below defines the key geometric parameters and material properties involved in such an assembly.

Shrink Fit Nomenclature — Nomenclature for a shrink-fit assembly. (a) shows the separate hub (reinforcement) and shaft (coil) before assembly, defining the radial interference $\delta$ as the sum of the radial changes in the hub ($\delta_h$) and shaft ($\delta_s$). (b) shows the assembled system with the resulting interface pressure $P$ at the common radius $b$.

2. The Theory: Stresses in Thick-Walled Cylinders

To calculate anything about a shrink-fit, we must first understand how stresses are distributed within a thick-walled cylinder under pressure. Your solid mechanics background gives you the fundamentals of stress and strain; here, we will apply them to this specific cylindrical geometry. The governing relations are known as Lamé's equations.

These equations describe the radial stress ( $σ_{r}$ ) and tangential (hoop) stress ( $σ_{t}$ ) at any radius $r$ within the cylinder's wall:
$σ_{r} = A - \frac{B}{r ^{2}}$
$σ_{t} = A + \frac{B}{r ^{2}}$
where $A$ and $B$ are constants determined by applying the boundary conditions—namely, the pressures at the inner and outer surfaces of the cylinder.

For our purpose, the shrink-fit problem involves two separate thick-cylinder problems that are coupled at their interface:

The inner coil, treated as a solid or hollow cylinder subjected to an external pressure $P$ from the reinforcement.
The outer reinforcement, treated as a hollow cylinder subjected to an internal pressure $P$ from the coil.

The following video provides an excellent and detailed explanation of the theory of thick-walled cylinders and then uses it to derive the equations for shrink-fits. The derivation is exactly the kind of detailed analysis you appreciate.

Stress Analysis: Thick Walled Pressure Vessels, Press & Shrink Fits (4 of 17)

This video from 'CPPMechEngTutorials' offers a university-level lecture on stress analysis. We will use it to first understand the stress distribution in thick-walled cylinders and then to see the detailed derivation of the equations for shrink-fits.

Please watch the following segments: Thick-Walled Cylinders & Lamé's Equations (00:46:55 - 00:57:49): Focus on the introduction of radial stress and the presentation of the general Lamé equations for hoop and radial stress. Don't worry about memorizing them; focus on what they represent. Internal Pressure Case (00:57:49 - 01:05:23): This simplifies the equations for the case of a cylinder with only internal pressure, which is exactly how we model the outer reinforcement jacket. Introduction to Shrink-Fits (01:05:23 - 01:12:47): This section conceptually introduces the shrink-fit process and defines the key term: radial interference. Derivation of the Interference-Pressure Relationship (01:12:47 - 01:24:21): This is the most important part. Follow the derivation that links the geometric interference (\delta) to the resulting interface pressure (P). This is the core calculation for our lesson.

3. Calculating the Interference Fit

As derived in the video, the total radial interference $δ$ (the initial difference in radii before assembly) is the sum of the radial expansion of the reinforcement and the radial compression of the coil at their common interface. This interference is what generates the contact pressure $P$ .

The relationship between the pressure $P$ and the radial interference $δ$ at the interface radius $R$ is the central formula for shrink-fit design. For an assembly of two cylinders made of potentially different materials, the formula is:

$P = \frac{δ}{R [ \frac{1}{E _{o}} ( \frac{R _{o}^{2} + R ^{2}}{R _{o}^{2} - R ^{2}} + ν _{o} ) + \frac{1}{E _{i}} ( \frac{R ^{2} + R _{i}^{2}}{R ^{2} - R _{i}^{2}} - ν _{i} ) ]}$

Where:

$P$ is the interface pressure.
$δ$ is the radial interference.
$R$ is the nominal radius of the interface.
$R_{i}$ is the inner radius of the inner cylinder (for a solid coil, $R_{i} = 0$ ).
$R_{o}$ is the outer radius of the outer cylinder.
$E_{i}, ν_{i}$ are the Young's modulus and Poisson's ratio for the inner cylinder (coil).
$E_{o}, ν_{o}$ are the Young's modulus and Poisson's ratio for the outer cylinder (reinforcement).

If the inner cylinder is a solid shaft ( $R_{i} = 0$ ) and both components are made of the same material ( $E_{o} = E_{i} = E$ , $ν_{o} = ν_{i} = ν$ ), the formula simplifies significantly. The video you watched presents these simplified cases as well.

The document below provides the derivations in a written format, which can be useful for reference.

Thick Walled Cylinders

The document 'Thick Walled Cylinders' from Roymech provides a concise, step-by-step derivation of the interference fit equations. It's a good text-based companion to the video.

Review the 'Interference Fit' section. Focus on the final equations that relate the 'total interference' (u_t) to the interface pressure (p_f). Note that their notation is slightly different (u_t is our \delta, p_f is our P).

4. The Design Workflow

In a real-world scenario, you would work through the following steps to design a shrink-fit reinforcement:

Determine Target Pre-stress: Decide on the desired initial compressive hoop stress ( $σ_{θ, pre}$ ) at the outer radius of the coil. This is a crucial design choice, often aimed at keeping the conductor in compression throughout the pulse.
Calculate Required Interface Pressure ( $P$ ): Using Lamé's equations for the inner coil (subjected to external pressure $P$ ), find the pressure $P$ that will produce the target pre-stress $σ_{θ, pre}$ . For a solid inner cylinder ( $R_{i} = 0$ ), the hoop stress is uniform and equal to $- P$ . Wait, that's not right. The stress is not uniform. For a solid cylinder under external pressure P, the stresses are $σ_{r} = - P$ and $σ_{t} = - P$ everywhere. For a hollow cylinder from $R_{i}$ to $R$ , subject to external pressure $P$ , the hoop stress at $r = R$ is $σ_{t} (R) = - P \frac{2 R _{i}^{2}}{R ^{2} - R _{i}^{2}}$ . This is a complex dependency. Let's use the simplest and most common case for a coil: treat it as a solid cylinder. For a solid cylinder under external pressure $P$ , the stresses are uniform throughout: $σ_{r} = - P$ and $σ_{t} = - P$ . So, the required interface pressure is simply $P = - σ_{θ, pre}$ .
Calculate Required Radial Interference ( $δ$ ): With the required pressure $P$ known, use the main formula from the previous section to calculate the necessary radial interference $δ$ . This is a key manufacturing specification.
Check Reinforcement Stresses: Calculate the hoop stress in the outer reinforcement jacket caused by the assembly pressure $P$ . The maximum hoop stress occurs at its inner surface ( $r = R$ ) and is given by:
$σ_{θ, reinforcement} (R) = P (\frac{R _{o}^{2} + R ^{2}}{R _{o}^{2} - R ^{2}})$
You must verify that this stress is safely below the yield strength of the reinforcement material.

Test your understanding!

Let's design a shrink-fit for a small test coil.

Coil: Solid copper, treated as a solid cylinder. Outer radius $R = 25$ mm.
- $E_{c} = 120$ GPa, $ν_{c} = 0.34$ .
Reinforcement: High-strength steel jacket. Outer radius $R_{o} = 50$ mm.
- $E_{s} = 200$ GPa, $ν_{s} = 0.3$ , Yield Strength $σ_{y, s} = 1500$ MPa.
Design Goal: Achieve a pre-compression hoop stress of $σ_{θ, pre} = - 300$ MPa in the copper coil.

Follow the design workflow:

What is the required interface pressure $P$ ?
Using the full formula, calculate the required radial interference $δ$ in micrometers ( $μ m$ ).
Calculate the maximum hoop stress in the steel jacket and check if it's safe.

Show answer

Interface Pressure $P$ :
For a solid inner cylinder, the hoop stress is uniform and $σ_{t} = - P$ .
To get $σ_{θ, pre} = - 300$ MPa, we need an interface pressure of $P = 300$ MPa.
Radial Interference $δ$ :
We rearrange the main formula to solve for $δ = P \times R \times [...bracketed term...]$ .
Let's calculate the two parts of the bracketed term:
- Outer Jacket (steel):
  $\frac{1}{E _{o}} (\frac{R _{o}^{2} + R ^{2}}{R _{o}^{2} - R ^{2}} + ν_{o}) = \frac{1}{200 \times 1 0 ^{9}} (\frac{5 0 ^{2} + 2 5 ^{2}}{5 0 ^{2} - 2 5 ^{2}} + 0.3) = \frac{1}{200 \times 1 0 ^{9}} (\frac{3125}{1875} + 0.3) = \frac{1.967}{200 \times 1 0 ^{9}} = 9.833 \times 1 0^{- 12} Pa^{- 1}$
- Inner Coil (copper, solid, so $R_{i} = 0$ ):
  $\frac{1}{E _{i}} (\frac{R ^{2} + R _{i}^{2}}{R ^{2} - R _{i}^{2}} - ν_{i}) = \frac{1}{120 \times 1 0 ^{9}} (\frac{2 5 ^{2} + 0 ^{2}}{2 5 ^{2} - 0 ^{2}} - 0.34) = \frac{1 - 0.34}{120 \times 1 0 ^{9}} = \frac{0.66}{120 \times 1 0 ^{9}} = 5.5 \times 1 0^{- 12} Pa^{- 1}$
- Total Interference $δ$ :
  $δ = P \cdot R \cdot (9.833 \times 1 0^{- 12} + 5.5 \times 1 0^{- 12}) = (300 \times 1 0^{6} Pa) \cdot (0.025 m) \cdot (15.333 \times 1 0^{- 12} Pa^{- 1})$
  $δ \approx 1.15 \times 1 0^{- 4} m = 115 μ m$
  The inner radius of the steel jacket must be manufactured to be about 115 $μ$ m smaller than the outer radius of the copper coil.
Check Reinforcement Stress:
The maximum hoop stress in the steel jacket is at its inner radius $r = R$ :
$σ_{θ, steel} (R) = P (\frac{R _{o}^{2} + R ^{2}}{R _{o}^{2} - R ^{2}}) = 300 MPa (\frac{5 0 ^{2} + 2 5 ^{2}}{5 0 ^{2} - 2 5 ^{2}}) = 300 MPa \times 1.667 = 500 MPa$
The calculated stress of 500 MPa is well below the steel's yield strength of 1500 MPa. The design is safe.

Conclusion

Today we have quantified the mechanics of one of the most important techniques in high-field magnet construction. Shrink-fitting is not just about assembly; it's a deliberate method of engineering the initial stress state of a coil to counteract the violent forces of a high-field pulse.

Key Takeaways:

Shrink-fitting a reinforcement jacket onto a coil creates a beneficial compressive pre-stress in the conductor.
This process is governed by the mechanics of thick-walled cylinders, described by Lamé's equations.
The radial interference—the initial geometric mismatch—is the key design parameter that controls the interface pressure.
We can calculate the required interference to achieve a target pre-stress, while also ensuring the reinforcement itself is not over-stressed during assembly.

Next Lesson Preview:
So far, we have focused entirely on the forces and stresses in the radial and tangential directions. We've tackled the "hoop stress" that tries to make the coil explode outwards. However, the Lorentz force $J \times B$ is a 3D vector field. There are also powerful axial forces that try to crush the coil from its ends, particularly in multi-layer coils. In our next lesson, we will learn how to calculate the axial electromagnetic forces between different sections of a multi-turn coil, which is the first step in designing structures to contain them.

Recommended resources

1. The Principle of Shrink-Fit Reinforcement

2. The Theory: Stresses in Thick-Walled Cylinders

3. Calculating the Interference Fit

4. The Design Workflow

Conclusion